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\newcommand{\ws}{Web-Service}
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\newcommand{\hbugs}{H-Bugs}
\newcommand{\helm}{HELM}

\title{Brokers and Web-Services for Automatic Deduction: a Case Study}

\author{Claudio Sacerdoti Coen \and Stefano Zacchiroli}

\institute{
  Department of Computer Science\\
  University of Bologna\\
  Via di Mura Anteo Zamboni 7, 40127 Bologna, ITALY\\
  \email{sacerdot@cs.unibo.it}
  \and
  Department of Computer Science\\
  \'Ecole Normale Sup\'erieure\\
  45, Rue d'Ulm, F-75230 Paris Cedex 05, FRANCE\\
  \email{zack@cs.unibo.it}
}

\date{ }

\begin{document}
\sloppy
\maketitle

\begin{abstract}
  We present a planning broker and several Web-Services for automatic deduction.
  Each Web-Service implements one of the tactics usually available in an
  interactive proof-assistant. When the broker is submitted a "proof status" (an
  unfinished proof tree and a focus on an open goal) it dispatches the proof to
  the Web-Services, collects the successfull results, and send them back to the
  client as "hints" as soon as they are available.
  
  In our experience this architecture turns out to be helpful both for
  experienced users (who can take benefit of distributing heavy computations)
  and beginners (who can learn from it).
\end{abstract}

\section{Introduction}
  The \ws\ approach at software development seems to be a working solution for
  getting rid of a wide range of incompatibilities between communicating
  software applications. W3C's efforts in standardizing related technologies
  grant longevity and implementations availability for frameworks based on \wss\
  for information exchange. As a direct conseguence, the number of such
  frameworks in increasing and the World Wide Web is moving from a disorganized
  repository of human-understandable HTML documents to a disorganized repository
  of applications working on machine-understandable XML documents both for input
  and output.
  
  This lack of organization along with the huge amount of documents available,
  have posed not a few problems for data location. If this limitation could be
  overcome using heuristic-based search engine for HTML pages, a satisfying
  solution has yet to be found for \ws\ location \todo{location non mi piace, ma
  "localization" e' anche peggio, "retrieval"?}. A promising architecture seems
  to be the use of \emph{brokers}, which access point (usually an URI) is known
  by the potential clients or by \emph{registries}, to which the \wss\ subscribe
  to publish their functionalities.

  Since the \emph{Declaration of Linz}, the MONET Consortium \cite{MONET} is
  working on the development of a framework aimed at providing a set of software
  tools for the advertisement and discovering of mathematical web services. This
  framework turns out to be strongly based on both \wss\ and brokers.

  Several examples of CAS (Computer Algebra System) and Theorem Prover related
  \wss\ have been shown to be useful and to fit well in the MONET Architecture
  \todo{citarne qualcuno: CSC???}. On the other hand \todo{troppo informale?}
  the set of \wss\ related to Proof Assistants tends to be ... rather ...
  empty!\todo{gia' mi immagino il tuo commento: BUHM! :-)}

  Despite of that the proof assistant case seems to be well suited to
  investigate the usage of many different mathematical \wss. Indeed: most proof
  assistants are still based on non-client/server architectures, are
  application-centric instead of document-centric, offer a scarce level of
  automation leaving entirely to the user the choice of which macro (usually
  called \emph{tactic}) to use in order to make progress in a proof.

  The average proof assistant can be, for example, a client of a \ws\
  interfacing a specific or generic purpose theorem prover, or a client of a
  \ws\ interfacing a CAS to simplify expressions in a particular mathematical
  domain.

  The Omega Ants project \todo{date, autori e paperi}, developed before the
  \emph{\ws\ era}, enrich the Omega \cite{Omega} proof assistant with an
  architecture able to distribute subparts of a proof to agents (Ants)
  implemented as separate processes running on the same hosts of the proof
  assistant \todo{controllare ...}.

  In this paper we present an architecture, namely \hbugs\, implementing a
  \emph{suggestion engine} for the proof assistant developed on behalf of the
  \helm\ project. This architecture is based on a set of \wss\ and a broker (a
  \ws\ itself) and is able to distribute a proof status from a client to several
  \emph{tutor}, each of them can try to make progress in the proof and, in case
  of success, notify the user sending him an \emph{hint}. Both the broker and
  the tutors are instances of the homonymous entities of the MONET framework.

% \section{Introduction}
% Since the development of the first proof-assistants based on the
% Curry-Howard isomorphism, it became clear that directly writing
% lambda-terms (henceforth called simply terms) was a difficult, repetitive,
% time-expensive and error prone activity; hence the introduction of
% meta-languages to describe procedures that are able to automatically
% generate the low-level terms.
% Nowadays, almost all the proof-assistants
% using terms description of proofs have many levels of abstractions, i.e.
% meta-languages, to create the terms, with the only remarkable exception
% of the ALF family \cite{ALF} in which terms are still directly written
% without any over-standing level of abstraction.
% 
% In particular, there are
% usually at least two levels, that of tactics and that of the language in
% which the whole system and hence the tactics are written; once the tactics
% are implemented, to do a proof the user enters to the system a sequence of
% tactics, called a script; the script is then executed to create the term
% encoding of the proof, which is type-checked by the kernel of the
% proof-assistant. Writing a script interactively is much simpler than
% writing the term by hands; once written, though, it becomes impossible
% to understand a script without replaying it interactively. For this reason,
% we can hardly speak of the language of tactics as a high level language,
% even if it is ``compiled'' to the language of terms.
% 
% To avoid confusion, in the rest of this paper we will
% avoid the use of the term ``proof'', using only ``script'' and ``term'';
% moreover, we will avoid also the terms ``proof-checking'' and ``type-checking'',
% replacing them with ``retyping'' to mean the type-checking of an already
% generated term; finally, we will use the term ``compiling'' to mean the
% execution of a script. Usually, compilation ends with
% the type-checking of the generated term; hence the choice of ``retyping'' for
% the type-checking of an already generated term.
% 
% A long term consequence of the introduction of tactics has been the
% progressive lowering of interest in terms. In particular, users of
% modern proof-assistants as Coq \cite{Coq} may even ignore the existence of
% commands to show the terms generated by their scripts. These terms are
% usually very huge and quite unreadable, so they don't add any easily
% accessible information to the scripts. Hence implementors have loosed
% interest in terms too, as far as their size and compilation time are
% small enough to make the system response-time acceptable.
% When this is not the case, it is sometimes possible to trade space
% with time using reflexive tactics, in which the potentiality of complex
% type-systems to speak about themselves are exploited: reflexive tactics
% usually leads to a polynomial asympotical reduction in the size of the
% terms, at the cost of an increased reduction time during retyping and
% compilation. Often, though, reflexive tactics could not be used; moreover,
% reflexive tactics suffer of the same implementative problems of other tactics
% and so could be implemented in such a way to create huger than
% needed terms, even if asymptotically smaller than the best ones created
% without reflection.
% 
% The low level of interest in terms of the implementors of
% tactics often leads to naive implementations (``If it works, it is OK'') and
% this to low-quality terms, which:
% \begin{enumerate}
%  \item are huger than needed because particular cases are not taken
%   into account during tactic development
%  \item require more time than needed for retyping due to
%   their size
%  \item are particularly unreadable because they don't
%   correspond to the ``natural'' way of writing the proof by hand
% \end{enumerate}
% To cope with the second problem, retyping is usually avoided allowing
% systems to reload saved terms without retyping them and using a
% checksum to ensure that the saved file has not been modified. This is
% perfectly reasonable accordingly to the traditional application-centric
% architecture of proof-assistants in which you have only one tool integrating
% all the functionalities and so you are free to use a proprietary format for data
% representation.
% 
% In the last months, though, an ever increasing number of people and projects
% (see, for example, HELM \cite{EHELM}, MathWeb \cite{MATHWEB} and
% Formavie \cite{FORMAVIE})
% have been interested to switch from the application-centric model to the
% newer content-centric one, in which information is stored in
% standard formats (that is, XML based) to allow different applications to work
% on the same set of data. As a consequence, term size really becomes an
% important issue, due to the redundancy of standard formats, and retyping
% is needed because the applications can not trust each other, hence needing
% retyping and making retyping time critical.
% Moreover, as showed by Yann Coscoy in its PhD.
% thesis \cite{YANNTHESIS} or by the Alfa interface to the Agda system
% \cite{ALFA}, it is perfectly reasonable and also feasible to try
% to produce descriptions in natural languages of formal proofs encoded as
% terms.
% This approach, combined with the further possibility of applying the usual
% two-dimensional mathematic notation to the formulas that appears in the
% terms, is being followed by projects HELM \cite{HELM}, PCOQ \cite{PCOQ} and
% MathWeb \cite{MATHWEB} with promising results. It must be understood, though,
% that the quality (in terms of naturality and readability) of this kind of
% proofs rendering heavily depends on the quality of terms, making also
% the third characteristic of low-quality terms a critical issue.
% 
% A totally different scenario in which term size and retyping time are
% critical is the one introduced by Necula and Lee \cite{Necula} under
% the name Proof Carrying Code (PCC). PCC is a technique that can be used
% for safe execution of untrusted code. In a typical instance of PCC, a code
% receiver establishes a set of safety rules that guarantee safe behavior
% of programs, and the code producer creates a formal safety proof that
% proves, for the untrusted code, adherence to the safety rules. Then, the
% proof is transmitted to the receiver together with the code and it is
% retyped before code execution. While very compact representation of the
% terms, highly based on type-inference and unification, could be used
% to reduce the size and retyping time \cite{Necula2}, designing proof-assistants
% to produce terms characterized by an high level of quality is still necessary.
% 
% In the next section we introduce a particular class of metrics for
% tactics evaluation. In section \ref{equivalence} we consider the
% notion of tactics equivalence and we describe one of the bad habits
% of tactics implementors, which is overkilling; we also provide and analyze
% a simple example of overkilling tactic. In the last section we describe
% a concrete experience of fixing overkilling in the implementation of a
% reflexive tactic in system Coq and we analyze the gain with respect to
% term-size, retyping time and term readability.
% 
% \section{From Metrics for Terms Evaluation to Metrics for Tactics Evaluation}
% The aim of this section is to show how metrics for term evaluation could
% induce metrics for tactic evaluation. Roughly speaking, this allows us to
% valuate tactics in terms of the quality of the terms produced. Even if
% we think that these kinds of metrics are interesting and worth studying,
% it must be understood that many other valuable forms of metrics could be
% defined on tactics, depending on what we are interested in. For example,
% we could be interested on compilation time, that is the sum of the time
% required to generate the term and the retyping time for it. Clearly, only
% the second component could be measured with a term metric and a good
% implementation of a tactic with respect to the metric considered could
% effectively decide to sacrifice term quality (and hence retyping time) to
% minimize the time spent to generate the term. The situation is very close to
% the one already encountered in compilers implementation, where there is
% always a compromise, usually user configurable, between minimizing compiling
% time and maximizing code quality.
% 
% The section is organized as follows: first we recall the definition of
% tactic and we introduce metrics on terms; then we give the definition
% of some metrics induced by term metrics on tactics.
% 
% \begin{definition}[Of tactic]
% We define a \emph{tactic} as a function that, given a goal $G$ (that is, a local
% context plus a type to inhabit) that satisfies a list of assumptions
% (preconditions) $P$,
% returns a couple $(L,t)$ where
% $L = {L_1,\ldots,L_n}$ is a (possible empty) list of proof-obligations
% (i.e. goals) and $t$ is a function that, given a list $l = {l_1,\ldots,l_n}$
% of terms such that $l_i$ inhabits\footnote{We say, with a small abuse of
% language, that a term $t$ inhabits a goal $G=(\Gamma,T)$ when $t$ is of type
% $T$ in the context $\Gamma$.} $L_i$ for each $i$ in ${1,\ldots,n}$, returns a
% term $t(l)$ inhabiting $G$.
% \end{definition}
% 
% % You can get another looser definition of tactic just saying that a tactic
% % is a partial function instead than a total one. In this paper when referring
% % to tactics we always refer to the first definition.
% 
% \begin{definition}[Of term metric]
% For any goal $G$, a term metric $\mu_G$ is any function in
% $\mathbb{N}^{\{t/t\;\mbox{inhabits}\;G\}}$.
% Two important class of term metrics are functional metrics and monotone
% metrics:
% \begin{enumerate}
% \item {\bf Functional metrics:} a metric $\mu_G$ is \emph{functional}
%  if for each term context (=term with an hole) $C[]$ and for all terms
%  $t_1$,$t_2$ if $\mu_G(t_1)=\mu_G(t_2)$ then $\mu_G(C[t_1])=\mu_G(C[t_2])$.
%  An equivalent definition is that a metric $\mu_G$ is functional
%  if for each term context $C[]$ the function
%  $\lambda t.\mu_G(C[\mu_G^{-1}(t)])$ is well defined.
% \item {\bf Monotone metrics:} a metric $\mu_G$ is \emph{monotone} if for
%  each term context $C[]$ and for all terms $t_1$,$t_2$ if
%  $\mu_G(t_1)\le\mu_G(t_2)$ then $\mu_G(C[t_1])\le\mu_G(C[t_2])$.
%  An equivalent definition is that a metric $\mu_G$ is functional if
%  for each term context $C[]$ the function
%  $\lambda t.\mu_G(C[\mu_G^{-1}(t)])$ is well defined and monotone.
% \end{enumerate}
% 
% % Vecchie definizioni
% %\begin{enumerate}
% %\item {\bf Monotony:} a metric $\mu_G$ is \emph{monotone} if for each term
%  %context (=term with an hole) $C[]$ and for all terms $t_1$,$t_2$ if
%  %$\mu_G(t_1)\le\mu_G(t_2)$ then $\mu_G(C[t_1])\le\mu_G(C[t_2])$
% %\item {\bf Strict monotony:} a monotone metric $\mu_G$ is \emph{strictly
%  %monotone} if for each term context $C[]$ and
%  %for all terms $t_1$,$t_2$ if $\mu_G(t_1)=\mu_G(t_2)$ then
%  %$\mu_G(C[t_1])=\mu_G(C[t_2])$
% %\end{enumerate}
% \end{definition}
% 
% Typical examples of term metrics are the size of a term, the time required
% to retype it or even an estimate of its ``naturality'' (or simplicity) to be
% defined somehow; the first two are also examples of monotone metrics
% and the third one could probably be defined as to be. So, in the rest
% of this paper, we will restrict to monotone metrics, even if the
% following definitions also work with weaker properties for general metrics.
% Here, however, we are not interested in defining such metrics, but in showing
% how they naturally induce metrics for tactics.
% 
% Once a term metric is chosen, we get the notion of a best term (not unique!)
% inhabiting a goal:
% \begin{definition}[Of best terms inhabiting a goal]
% the term $t$ inhabiting a goal $G$ is said to be a best term inhabiting $G$
% w.r.t. the metric $\mu_G$ when
% $\mu_G(t) = min\{\mu_G(t') / t'\;\mbox{inhabits}\;G\}$.
% \end{definition}
% 
% Using the previous notion, we can confront the behavior of two tactics
% on the same goal:
% 
% \begin{definition}
% Let $\tau_1$ and $\tau_2$ be two tactics both applyable to a goal $G$
% such that $\tau_1(G) = (L_1,t_1)$ and $\tau_2(G) = (L_2,t_2)$.
% We say that $\tau_1$ is better or equal than $\tau_2$ for the goal $G$
% with respect to $\mu_G$ if, for all $l_1$ and $l_2$ lists of best terms
% inhabiting respectively $L_1$ and $L_2$, $\mu_G(t_1(l_1)) \le \mu_G(t_2(l_2))$
% holds.
% \end{definition}
% Note that confronting in this way ``equivalent'' tactics
% (whose definition is precised in the next section) gives us information
% on which implementation is better; doing the same thing on tactics that
% are not equivalent, instead, gives us information about what tactic to
% apply to obtain the best proof.
% 
% A (functional) metric to confront two tactics only on a particular goal
% has the nice property to be a total order, but is quite useless. Hence,
% we will now
% define a bunch of different tactic metrics induced by term metrics
% that can be used to confront the behavior of tactics when applied to
% a generic goal. Some of them will be \emph{deterministic} partial orders;
% others will be total orders, but will provide only a \emph{probabilistic}
% estimate of the behavior. Both kinds of metrics are useful in practice
% when rating tactics implementation.
% \begin{definition}[Of locally deterministic better or equal tactic]
% $\tau_1$ is a locally deterministic (or locally
% uniform) better or equal tactic
% than $\tau_2$ w.r.t. $\mu$ (and in that case we write $\tau_1 \le_{\mu} \tau_2$
% or simply $\tau_1 \le \tau_2$), when
% for all goals $G$ satisfying the preconditions of both tactics we have that
% $\tau_1$ is better or equal than $\tau_2$ w.r.t. the metric $\mu_G$.
% \end{definition}
% 
% \begin{definition}[Of locally deterministic better tactic]
% $\tau_1$ is a locally deterministic (or locally uniform)
% better tactic than $\tau_2$ 
% w.r.t. $\mu$ (and in that case we write $\tau_1 <_{\mu} \tau_2$
% or simply $\tau_1 < \tau_2$), when $\tau_1 \le_{\mu} \tau_2$ and
% exists a goal $G$ satisfying the preconditions of both tactics such that
% $\tau_1$ is better (but not equal!) than $\tau_2$ w.r.t. the metric $\mu_G$.
% \end{definition}
% 
% \begin{definition}[Of locally probabilistic better or equal tactic of a factor K]
% $\tau_1$ is said to be a tactic locally probabilistic better or equal of a
% factor $0.5 \le K \le 1$ than $\tau_2$ w.r.t. $\mu$ and a particular expected
% goals distribution when the probability of having $\tau_1$ better or equal than
% $\tau_2$ w.r.t. the metric $\mu_G$ is greater or equal to $K$ when $G$ is
% chosen randomly according to the distribution.
% \end{definition}
% The set of terms being discrete, you can note that a deterministically better
% or equal tactic is a tactic probabilistically better or equal of a factor 1.
% 
% 
% To end this section, we can remark the strong dependence of the $\le$
% relation on the choice of metric $\mu$, so that it is easy to find
% two metrics $\mu_1,\mu_2$ such that $\tau_1 <_{\mu_1} \tau_2$ and
% $\tau_2 <_{\mu_2} \tau_1$. Luckily, tough, the main interesting metrics,
% term size, retyping time and naturality, are in practice highly correlated,
% though the correlation of the third one with the previous two could be a
% bit surprising. So, in the following section, we
% will not state what is the chosen term metric; you may think as
% any of them or even at some kind of weighted mean.
% 
% 
% \section{Equivalent Tactics and Overkilling}
% \label{equivalence}
% We are now interested in using the metrics defined in the previous section
% to confront tactics implementation. Before doing so, though, we have to
% identify what we consider to be different implementations of the
% same tactic. Our approach consists in identifying every implementation
% with the tactic it implements and then defining appropriate notions of
% equivalence for tactics: two equivalent tactics will then be considered
% as equivalent implementations and will be confronted using metrics.
% 
% Defining two tactics as equivalent when they can solve exactly the same
% set of goals generating the same set of proof-obligations seems quite natural,
% but is highly unsatisfactory if not completely wrong. The reason is that, for
% equivalent tactics, we would like to have the \emph{property of substitutivity},
% that is substituting a tactic for an equivalent one in a script should give
% back an error-free script\footnote{A weaker notion of substitutivity is that
% substituting the term generated by a tactic for the term generated by an
% equivalent one in a generic well-typed term should always give back a
% well-typed term.}. In logical frameworks with dependent types,
% without proof-irrelevance and with universes as CIC \cite{Werner} though,
% it is possible for a term to inspect the term of a previous proof, behaving
% in a different way, for example, if the constructive proof of a conjunction
% is made proving the left or right side.
% So, two tactics, equivalent w.r.t.  the previous
% definition, that prove $A \vee A$ having at their disposal an
% hypothesis $A$ proving the first one the left and the second one
% the right part of the conjunction, could not be substituted one for the
% other if a subsequent term inspects the form of the generated proof.
% 
% Put in another way, it seems quite reasonable to derive equivalence for
% tactics from the definition of an underlying equivalence for terms.
% The simplest form of such an equivalence relation is convertibility
% (up to proof-irrelevance) of closed terms and this is the relation
% we will use in this section and the following one. In particular,
% we will restrict ourselves to CIC and hence to
% $\beta\delta\iota$-convertibility\footnote{The Coq proof-assistant
% introduces the notion of \emph{opaque} and \emph{transparent} terms,
% differing only for the possibility of being inspected. Because the
% user could change the opacity status at any time, the notion of
% convertibility we must conservatively choose for the terms of Coq is
% $\beta\delta\iota$-convertibility after having set all the definitions
% as transparent.}.
% Convertibility, though, is a too restrictive notion that does not take
% in account, for example, commuting conversions. Looking for more suitable
% notions of equivalence is our main open issue for future work.
% 
% \begin{definition}[Of terms closed in a local environment]
% A term $t$ is \emph{closed} in a local environment $\Gamma$ when $\Gamma$
% is defined on any free variable of $t$.
% \end{definition}
% 
% \begin{definition}[Of equivalent tactics]
% We define two tactics $\tau_1$ and $\tau_2$ to be equivalent (and
% we write $\tau_1 \approx \tau_2$) when for each goal $G = (\Gamma,T)$ and for
% each list of terms closed in $\Gamma$ and inhabiting the proof-obligations
% generated respectively by $\tau_1$ and $\tau_2$, we have that the result terms
% produced by $\tau_1$ and $\tau_2$ are $\beta\delta\iota$-convertible.
% \end{definition}
% 
% Once we have the definition of equivalent tactics, we can use metrics,
% either deterministic or probabilistic, to confront them. In particular,
% in the rest of this section and in the following one we will focus on the
% notion of deterministically overkilling tactic, defined as follows:
% 
% \begin{definition}[Of overkilling tactics]
% A tactic $\tau_1$ is (deterministically) overkilling w.r.t. a metric
% $\mu$ when there exists another tactic $\tau_2$ such that
% $\tau_1 \approx \tau_2$ and $\tau_2 <_\mu \tau_1$.
% \end{definition}
% 
% Fixing an overkilling tactic $\tau_1$ means replacing it with the
% tactic $\tau_2$ which is the witness of $\tau_1$ being overkilling.
% Note that the fixed tactic could still be overkilling.
% 
% The name overkilling has been chosen because most of the time overkilling
% tactics are tactics that do not consider special cases, following the general
% algorithm. While in computer science it is often a good design pattern to
% prefer general solutions to ad-hoc ones, this is not a silver bullet:
% an example comes another time from compiler technology, where
% ad-hoc cases, i.e. optimizations, are greatly valuable if not necessary.
% In our context, ad-hoc cases could be considered either as optimizations,
% or as applications of Occam's razor to proofs to keep the simplest one.
% 
% \subsection{A Simple Example of Overkilling Tactic}
% A first example of overkilling tactic in system Coq is
% \emph{Replace}, that works in this way: when the current goal
% is $G = (\Gamma,T)$, the
% tactic ``{\texttt Replace E$_1$ with E$_2$.}'' always produces a new
% principal proof-obligation $(\Gamma, T\{E_2/E_1\})$ and an auxiliary
% proof-obligation $(\Gamma, E_1=E_2)$ and uses the elimination scheme
% of equality on the term $E_1 = E_2$ and the two terms that inhabit the
% obligations to prove the current goal.
% 
% To show that this tactic is overkilling, we will provide an example
% in which the tactic fails to find the best term, we will
% propose a different implementation that produces the best term and we will
% show the equivalence with the actual one.
% 
% The example consists in applying the tactic in the case in which $E_1$ is
% convertible to $E_2$: the tactic proposes to the user the two
% proof-obligations and then builds the term as described above. We claim
% that the term inhabiting the principal proof-obligation also inhabits
% the goal and, used as the generated term, is surely smaller
% and quicker to retype than the one that is generated in the
% implementation; moreover, it
% is also as natural as the previous one, in the sense that the apparently
% lost information has simply become implicit in the reduction and could
% be easily rediscovered using type-inference algorithms as the one described
% in Coscoy's thesis \cite{YANNTHESIS}. So, a new implementation could
% simply recognize this special case and generate the better term.
% 
% We will now show that the terms provided by the two implementations are
% $\beta\delta\iota$-convertible.
% Each closed terms in $\beta\delta\iota$-normal form
% inhabiting the proof that $E_1$ is equal to $E_2$ is equal to the only
% constructor of equality applied to a term convertible to the type of
% $E_1$ and to another term convertible to $E_1$; hence, once the principle
% of elimination of equality is applied to this term, we can first apply
% $\beta$-reduction and then $\iota$-reduction to obtain the term inhabiting
% the principal proof-obligation in which $E_1$ has been replaced by $E_2$.
% Since $E_1$ and $E_2$ are $\beta\delta\iota$-convertible by hypothesis and
% for the congruence properties of convertibility in CIC, we have that the
% generated term is $\beta\delta\iota$-convertible to the one inhabiting the
% principal proof-obligation.
% 
% %The simplest example is when $E_1$ and $E_2$ are syntactically equal:
% %in this case the principal obligation proposed is identical to the current goal
% %and the equality elimination introduced could be replaced by the proof of
% %this obligation, leading to a smaller, quicker to retype and also more
% %natural\footnote{Even if naturality is in some way a subjective metric,
% %who would dare say that a proof in which you rewrite an expression with
% %itself is natural?} term. To show that handling this special case in this
% %way is equivalent to the current solution, we have to show that the two
% %terms are $\beta\delta\iota$-convertible when so are the subterms inhabiting
% %the obligations.
% %The only closed term in $\beta\delta\iota$-normal form
% %inhabiting the proof that $E_1$ is equal to $E_1$ is the only constructor
% %of equality applied to the type of $E_1$ and to $E_1$; hence we can
% %first apply $\beta$-reduction and then $\iota$-reduction to obtain
% %the second proof in which $E_1$ has been replaced by $E_1$, i.e. the
% %second proof.
% 
% %So, for this example, the simple fix to avoid overkilling is checking
% %if $E_1$ and $E_2$ are syntactically equal and in case omitting the elimination;
% %curiously, this is done in Coq in the implementation of a very similar tactic
% %called \emph{Rewriting}.
% 
% 
% %A second, more involved, example is when $E_1$ and $E_2$ are not
% %syntactically equal, but $E_1$ reduces to $E_2$. Even in this case
% %the elimination could simply be avoided because the typing rules of
% %the logical system ensures us that a term that inhabits the principal
% %obligation also inhabits the current goal: the proof is equivalent to the
% %previous one, but for the last step in which $E_2$ is actually substituted for
% %$E_1$; by hypothesis, though, the two terms are $\beta\delta\iota$-convertible
% %and hence the thesis.
% 
% %Surely smaller and faster to retype, the new term is also as natural
% %as the previous one, in the sense that the apparently lost information has
% %simply become implicit in the reduction and could be easily rediscovered
% %using type-inference algorithms as the one described in chapter ???
% %of Coscoy's thesis \ref{YANNTHESIS}.
% 
% This example may seem quite stupid because, if the user is already able to
% prove the principal proof-obligation and because this new goal is totally
% equivalent to the original one, the user could simply redo the same steps
% without applying the rewriting at all. Most of the time, though, the
% convertibility of the two terms could be really complex to understand,
% greatly depending on the exact definitions given; indeed, the user could
% often be completely unaware of the convertibility of the two terms. Moreover,
% even in the cases in which the user understands the convertibility, the
% tactic has the important effect of changing the form of the current
% goal in order to simplify the task of completing the proof, which is the reason
% for the user to apply it.
% 
% ~\\
% 
% The previous example shows only a very small improvement in the produced
% term and could make you wonder if the effort of fixing overkilling and
% more in general if putting more attention to terms when implementing
% tactics is really worth the trouble. In the next section we describe as
% another example a concrete experience of fixing a complex reflexive tactic
% in system Coq that has lead to really significant improvements in term
% size, retyping time and naturality.
% 
% \section{Fixing Overkilling: a Concrete Experience}
% Coq provides a reflexive tactic called \emph{Ring} to do associative-commutative
% rewriting in ring and semi-ring structures. The usual usage is,
% given the goal $E_1 = E_2$ where $E_1$ and $E_2$ are two expressions defined
% on the ring-structure, to prove the goal reducing it to proving
% $E'_1 = E'_2$ where $E'_i$ is the normal form of $E_i$. In fact, once
% obtained the goal $E'_1 = E'_2$, the tactic also tries to apply simple
% heuristics to automatically solve the goal.
% 
% The actual implementation of the tactic by reflexion is quite complex and
% is described in \cite{Ring}. The main idea is described in Fig. \ref{ring1}:
% first of all, an inductive data type to describe abstract polynomial is
% made available. On this abstract polynomial, using well-founded recursion in
% Coq, a normalization function named $apolynomial\_normalize$ is defined;
% for technical reasons, the abstract data type of normalized polynomials
% is different from the one of un-normalized polynomials. Then, two
% interpretation functions, named $interp\_ap$ and $interp\_sacs$ are given
% to map the two forms of abstract polynomials to the concrete one. Finally,
% a theorem named $apolynomial\_normalize\_ok$ stating the equality of
% the interpretation of an abstract polynomial and the interpretation of
% its normal form is defined in Coq using well-founded induction. The above
% machinery could be used in this way: to prove that $E^I$ is equal to its
% normal form $E^{IV}$, the tactic computes an abstract polynomial $E^{II}$ that,
% once interpreted, reduces to $E^{I}$, and such that the interpretation
% of $E^{III} = (apolynomial\_normalize E^{II})$ could be shown to be equal
% to $E^{IV}$ applying $apolynomial\_normalize\_ok$.
% 
% %such that $(interp\_ap\;E^{II})$ could be proved (theorem
% %$apolynomial\_normalize\_ok$) in Coq to be equal to
% %$(interp\_sacs\;E^{III})$ where
% %$E^{III} = (apolynomial\_normalize E^{II})$ is another abstract polynomial
% %in normal form and the three functions $interp\_ap$, $interp\_sacs$ and
% %$apolynomial\_normalize$ could all be defined inside Coq using well-founded
% %recursion/induction on the abstract polynomial definition.
% 
% % \myincludegraphics{ring1}{t}{12cm}{Reflexion in Ring}{Reflexion in Ring}
% % \myincludegraphics{ring2}{t}{10cm}{Ring implementation (first half)}{Ring implementation (first half)}
% 
% In Fig. \ref{ring2} the first half of the steps taken
% by the Ring tactic to prove $E_1 = E_2$ are shown\footnote{Here $E_1$
% stands for $E^I$.}.
% The first step is replacing $E_1 = E_2$ with $(interp\_ap\;E^{II}_1) = E_2$,
% justifying the rewriting using the only one constructor of equality due to
% the $\beta\delta\iota$-convertibility of $(interp\_ap\;E^{II}_1)$ with $E_1$.
% The second one is replacing $(interp\_ap\;E^{II})$ with
% $(interp\_sacs\;E^{III})$, justifying the rewriting using
% $apolynomial\_normalize\_ok$.
% 
% Next, the two steps are done again on the left part of the equality,
% obtaining $(interp\_sacs\;E^{III}_1) = (interp\_sacs\;E^{III}_2)$,
% that is eventually solved trying simpler tactics as \emph{Reflexivity} or left
% to the user.
% 
% The tactic is clearly overkilling, at least due to the usage of rewriting for
% convertible terms. Let's consider as a simple example the session in Fig.
% \ref{session}:
% \begin{figure}[t]
% %\begin{verbatim}
% \mbox{\hspace{3cm}\tt Coq $<$ Goal ``0*0==0``.}\\
% \mbox{\hspace{3cm}\tt 1 subgoal}\\
% ~\\
% \mbox{\hspace{3cm}\tt ~~=================}\\
% \mbox{\hspace{3cm}\tt ~~~~``0*0 == 0``}\\
% ~\\
% \mbox{\hspace{3cm}\tt Unnamed\_thm $<$ Ring.}\\
% \mbox{\hspace{3cm}\tt Subtree proved!}
% %\end{verbatim}
% \caption{A Coq session.}
% \label{session}
% \end{figure}
% in Fig. \ref{before} the $\lambda$-term created by the
% original overkilling implementation of Ring is shown. Following the previous
% explanation, it should be easily understandable. In particular, the
% four rewritings are clearly visible as applications of $eqT\_ind$, as
% are the two applications of $apolynomial\_normalize\_ok$ and the
% three usage of reflexivity, i.e. the two applications of $refl\_eqT$
% to justify the rewritings on the left and right members of the equality
% and the one that ends the proof.
% 
% Let's start the analysis of overkilling in this implementation:
% \begin{description}
% \item[Always overkilling rewritings:] as already stated, four of the rewriting
%  steps are always overkilling because the rewritten term is convertible to
%  the original one due to the tactic implementation.
%  As proved in the previous section, all these rewritings could be simply
%  removed obtaining an equivalent tactic.
% \item[Overkilling rewritings due to members already normalized:] it may happen,
%  as in our example, that one (or even both) of the two members is already in
%  normal form. In this case the two rewriting steps for that member could be
%  simply removed obtaining an equivalent tactic as shown in the previous section.
% \item[Rewriting followed by reflexivity:] after having removed all
%  the overkilling rewritings, the general form of the $\lambda$-term produced
%  for $E_1 = E_2$ is the application of two rewritings ($E'_1$ for $E_1$ and
%  $E'_2$ for $E_2$), followed by a proof of $E'_1 = E'_2$. In many cases,
%  $E'_1$ and $E'_2$ are simply convertible and so the tactic finishes the
%  proof with an application of reflexivity to prove the equivalent goal
%  $E'_1 = E'_1$.
%  A smaller and also more natural solution is just to
%  rewrite $E'_1$ for $E_1$ and then proving $E'_1 = E_2$ applying the lemma
%  stating the symmetry of equality to the proof of $E_2 = E'_2$.
%  The equivalence to the original
%  tactic is trivial by $\beta\iota$-reduction because the lemma is proved
%  exactly doing the rewriting and then applying reflexivity:
%  $$
%  \begin{array}{l}
%  \lambda A:Type.\\
%  \hspace{0.2cm}\lambda x,y:A.\\
%  \hspace{0.4cm}\lambda H:(x==y).\\
%  \hspace{0.6cm}(eqT\_ind~A~x~ [x:A]a==x~(refl\_eqT~A~x)~y~H)
%  \end{array}
%  $$
% \end{description}
% In Fig. \ref{after} is shown the $\lambda$-term created by the same
% tactic after having fixed all the overkilling problems described above.
% 
% \begin{figure}
% \begin{verbatim}
% Unnamed_thm < Show Proof.
% LOC: 
% Subgoals
% Proof:
% (eqT_ind R
%   (interp_ap R Rplus Rmult ``1`` ``0`` Ropp (Empty_vm R)
%     (APmult AP0 AP0)) [r:R]``r == 0``
%   (eqT_ind R
%     (interp_sacs R Rplus Rmult ``1`` ``0`` Ropp (Empty_vm R)
%       Nil_varlist) [r:R]``r == 0``
%     (eqT_ind R
%       (interp_ap R Rplus Rmult ``1`` ``0`` Ropp (Empty_vm R) AP0)
%       [r:R]
%        ``(interp_sacs R Rplus Rmult 1 r Ropp (Empty_vm R) Nil_varlist)
%        == r``
%       (eqT_ind R
%         (interp_sacs R Rplus Rmult ``1`` ``0`` Ropp (Empty_vm R)
%           Nil_varlist)
%         [r:R]
%          ``(interp_sacs R Rplus Rmult 1 r Ropp (Empty_vm R)
%            Nil_varlist) == r`` (refl_eqT R ``0``)
%         (interp_ap R Rplus Rmult ``1`` ``0`` Ropp (Empty_vm R) AP0)
%         (apolynomial_normalize_ok R Rplus Rmult ``1`` ``0`` Ropp
%           [_,_:R]false (Empty_vm R) RTheory AP0)) ``0``
%       (refl_eqT R ``0``))
%     (interp_ap R Rplus Rmult ``1`` ``0`` Ropp (Empty_vm R)
%       (APmult AP0 AP0))
%     (apolynomial_normalize_ok R Rplus Rmult ``1`` ``0`` Ropp
%       [_,_:R]false (Empty_vm R) RTheory (APmult AP0 AP0))) ``0*0``
%   (refl_eqT R ``0*0``))
% \end{verbatim}
% \caption{The $\lambda$-term created by the original overkilling implementation}
% \label{before}
% \end{figure}
% 
% \begin{figure}
% \begin{verbatim}
% Unnamed_thm < Show Proof.
% LOC: 
% Subgoals
% Proof:
% (sym_eqT R ``0`` ``0*0``
%   (apolynomial_normalize_ok R Rplus Rmult ``1`` ``0`` Ropp
%     [_,_:R]false (Empty_vm R) RTheory (APmult AP0 AP0)))
% \end{verbatim}
% \caption{The $\lambda$-term created by the new implementation}
% \label{after}
% \end{figure}
% 
% \clearpage
% \subsection{A Quantitative Analysis of the Gain Obtained}
% Let's now try a quantitative analysis of the gain with respect to term size,
% retyping time and naturality, considering the two interesting cases of
% no member or only one member already in normal form\footnote{If the two
% members are already in normal form, the new implementation simply applies
% once the only constructor of the equality to one of the two members. The tactic
% is also implemented to do the same thing also when the two members are not yet
% in normal forms, but are already convertible. We omit this other improvement
% in our analysis.}.
% 
% \subsubsection{Term Size.}
% \paragraph{Terms metric definition:} given a term $t$, the metric $|.|$ associates to it its number of nodes $|t|$.
% \paragraph{Notation}: $|T|$ stands for the number of nodes in the actual parameters
%  given to $interp\_ap$, $interp\_sacs$ and $apolynomial\_normalize\_ok$ to
%  describe the concrete (semi)ring theory and the list of non-primitive
%  terms occurring in the goal to solve. In the example in figures \ref{before}
%  and \ref{after}, $|T|$ is the number of nodes in
%  \begin{texttt}[R Rplus Rmult ``1`` ``0`` Ropp (Empty\_vm R)]\end{texttt}.
%  $|R|$ stands for the number of nodes in the term which is the carrier
%  of the ring structure. In the same examples, $|R|$ is simply 1, i.e.
%  the number of nodes in \begin{texttt}R\end{texttt}.
% 
% \begin{displaymath}
% \begin{array}{l}
% \mbox{\bf Original version:}\\
% \begin{array}{ll}
% 1 + (|E^{II}_1| + |T| + 1) + |E_2| + |E_1| + &
% \mbox{(I rewriting Left)} \\
% 1 + |E_1| +  &
% \mbox{(justification)} \\
% 1 + (|E^{III}_1| + |T| + 1) + |E_2| + (|E^{II}_1| + |T| + 1) + &
% \mbox{(II rewriting Left)} \\
% (|E^{II}_1| + |T| + 1) + &
% \mbox{(justification)} \\
% 1 + (|E^{II}_2| + |T| + 1) + (|E^{III}_1| + |T| + 1) + |E_2| + &
% \mbox{(I rewriting Right)} \\
% 1 + |E_2| + &
% \mbox{(justification)} \\
% 1 + (|E^{III}_2| + |T| + 1) + (|E^{III}_1| + |T| + 1) + &
% \mbox{(II rewriting Right)} \\
% ~(|E^{II}_2| + |T| + 1) +& \\
% (|E^{II}_2| + |T| + 1) + &
% \mbox{(justification)} \\
% 1 + |E_1| = &
% \mbox{(reflexivity application)} \\
% \hline
% 4|E_1| + 2|E_2| + 3|E^{II}_1| + 3|E^{II}_2| + 3|E^{III}_1| + |E^{III}_2| +~ &
% \mbox{\bf Total number} \\
% ~10|T| + 17 &
% \end{array}
% \end{array}
% \end{displaymath}
% 
% \begin{displaymath}
% \begin{array}{l}
% \mbox{\bf New version, both members not in normal form:}\\
% \begin{array}{ll}
% 1 + |R| + |E_1| + |E'_2| + &
% \mbox{(Rewriting Right)} \\
% 1 + |T| + |E^{II}_2| + &
% \mbox{(justification)} \\
% 1 + |R| + |E'_2| + |E'_1| + |E_2| + &
% \mbox{(Symmetry application)} \\
% 1 + |T| + |E^{II}_1| = &
% \mbox{(justification)} \\
% \hline
% 2|E_1| + |E_2| + |E^{II}_1| + |E^{II}_2| + 2|E'_2| + 2|T| +~ &
% \mbox{\bf Total number} \\
% ~2|R| + 4  & \\
% ~ &
% \end{array}
% \end{array}
% \end{displaymath}
% \begin{displaymath}
% \begin{array}{l}
% \mbox{\bf New version, only the first member not in normal form:}\\
% \begin{array}{ll}
% 1 + |R| + |E_1| + |E'_2| + &
% \mbox{(Rewriting)} \\
% 1 + |T| + |E^{II}_2| = &
% \mbox{(justification)} \\
% \hline
% |E_1| + |E'_2| + |E^{II}_2| + |T| + |R| + 2~~~  &
% \mbox{\bf Total number} \\
% ~ &
% \end{array}
% \end{array}
% \end{displaymath}
% 
% While the overall space complexity of the terms generated by the new
% implementation is asymptotically equal to the one of the old implementation,
% all the constants involved are much smaller, but for the one of
% $E'_2$ (the two normal forms) that before was 0 and now is
% equal to 2. Is it possible to have goals for which the new implementation
% behaves worst than the old one? Unfortunately, yes. This happens when
% the size of the two normal forms $E'_1$ and $E'_2$ is greatly huger than
% ($E^{II}_1 + |T| + 1)$ and $(E^{II}_2 + |T| + 1)$. This happens when
% the number of occurrences of non-primitive terms is much higher than
% the number of non-primitive terms and the size of them is big. More
% formally, being $m$ the number of non-primitive terms, $d$ the average
% size and $n$ the number of occurrences, the new implementation creates bigger
% terms than the previous one if
% \begin{displaymath}
% n \log_2 m + m  d < n  d
% \end{displaymath}
% where the difference between the two members is great enough to hide the gain
% achieved lowering all the other constants.
% The logarithmic factor in the previous
% formula derives from the implementation of the map of variables to
% non-primitive terms as a tree and the representation of occurrences with
% the path inside the tree to retrieve the term.
% 
% To fix the problem, for each non-primitive term occurring more than once
% inside the normal forms, we can use a \emph{let \ldots in} local definition
% to bind it to a fresh identifier; then we replace every occurrence of
% the term inside the normal forms with the appropriate
% identifier\footnote{This has not yet been implemented in Coq.}.
% In this way, the above inequation becomes
% \begin{displaymath}
% n \log_2 m + m  d < n + m  d
% \end{displaymath}
% that is never satisfied.
% 
% Here it is important to stress how the latest problem was easily
% overlooked during the implementation and has been discovered
% only during the previous analysis, strengthening our belief in
% the importance of this kind of analysis for tactic implementations.
% 
% In the next two paragraphs we will consider only the new implementation
% with the above fixing.
% 
% \subsubsection{Retyping Time.}
% \paragraph{Terms metric definition:} given a term $t$, the metric $|.|$ associates to it the time $|t|$ required to retype it.
% 
% Due to lack of space, we will omit a detailed analysis as the one given
% for terms size. Nevertheless, we can observe that the retyping time required
% is surely smaller because all the type-checking operations required for
% the new implementation are already present in the old one, but for the
% type-checking of the two normal forms, that have fewer complexity
% than the type-checking of the two abstract normal forms, and the
% \emph{let \ldots in} definitions that have the same complexity of the
% type-checking of the variable map. Moreover, the quite expensive operation
% of computing the two normal forms is already done during proof construction.
% 
% %In the case in which both the members of the equality are not in normal form,
% %we can't expect a great improvement in retyping time but for very cheap
% %normalizations; this because retyping time is dominated by the factor due to
% %$apolynomial\_normalize$ that is present in both implementations and is
% %usually much higher than the factor due to the removed applications of
% %$interp\_ap$, $interp\_sacs$ and the eliminations of equality.
% 
% %The situation is very different if one of the two members is already
% %convertible to its normal form and the reduction involved in the normalization
% %is expensive. In this rather unlikely case, the new implementation
% %avoids the reduction at all, roughly halving the overall retyping time.
% 
% In section \ref{benchmarks} we present some benchmarks to give an idea of
% the real gain obtained.
% 
% \subsubsection{Naturality.}~\\~\\
% The idea behind the \emph{Ring} tactic is to be able to prove
% an equality showing that both members have the same normal form.
% This simply amounts to show that each member is equal to the same
% normal form, that is exactly what is done in the new implementation.
% Indeed, every step that belonged to the old implementation and has been
% changed or removed to fix overkilling used to lead to some unnatural step:
% \begin{enumerate}
% \item The fact that the normalization is not done on the concrete
%  representation, but passing through two abstract ones that are
%  interpreted on the concrete terms is an implementative detail that
%  was not hidden as much as possible as it should be.
% \item Normalizing a member of the equality that is already in normal form,
%  is illogical and so unnatural. Hence it should be avoided, but it was not.
% \item The natural way to show $A=B$ under the hypothesis $B=A$ is just
%  to use the symmetric property of equality. Instead, the old implementation
%  rewrote $B$ with $A$ using the hypothesis and proved the goal by reflexivity.
% \item Using local definitions (\emph{let \ldots in}) as abbreviations
%  rises the readability of the proof by shrinking its size removing
%  subexpressions that are not involved in the computation.
% \end{enumerate}
% 
% \subsection{Some Benchmarks}
% \label{benchmarks}To understand the actual gain in term size and retyping
% time on real-life examples, we have done some benchmarks on the whole set
% of theorems in the standard library of Coq that use the Ring tactic. The
% results are shown in table~\ref{benchs}.
% 
% Term size is the size of the disk dump of the terms. Re-typing time is the
% user time spent by Coq in proof-checking already parsed terms. The reduction
% of the terms size implies also a reduction in Coq parsing time, that is
% difficult to compute because Coq files do not hold single terms, but whole
% theories. Hence, the parsing time shown is really the user time spent by Coq
% to parse not only the terms on which we are interested, but also all the
% terms in their theories and the theories on which they depend. So, this last
% measure greatly under-estimates the actual gain.
% 
% Every benchmark has been repeated 100 times under different load conditions on
% a 600Mhz Pentium III bi-processor equipped with 256Mb RAM. The timings shown
% are mean values.
% 
% \begin{table}
% \begin{center}
% \begin{tabular}{|l|c|c|c|}
% \hline
%  & ~Term size~ & Re-typing time & Parsing time\\
% \hline
% Old implementation & 20.27Mb & 4.59s & 2.425s \\
% \hline
% New implementation & 12.99Mb & 2.94s & 2.210s \\
% \hline
% Percentage reduction & 35.74\% & 35.95\% & 8.87\% \\
% \hline
% \end{tabular}
% \end{center}
% \caption{Some benchmarks}
% \label{benchs}
% \end{table}
% 
% \section{Conclusions and Future Work}
% Naive ways of implementing tactics lead to low quality terms that are
% difficult to inspect and process. To improve the situation, we show
% how metrics defined for terms naturally induce metrics for tactics and
% tactics implementation and we advocate the usage of such metrics for
% tactics evaluation and implementation. In particular, metrics could
% be used to analyze the quality of an implementation or could be used at run
% time by a tactic to choose what is the best way to proceed.
% 
% To safely replace a tactic implementation with another one, it is important
% to define when two tactics are equivalent, i.e. generate equivalent terms.
% In this work, the equivalence relation chosen for terms has simply been 
% $\beta\delta\iota$-convertibility, that in many situations seems too strong.
% Hence, an important future work is the study of weaker forms of term
% equivalence and the equivalence relations they induce on tactics. In particular,
% it seems that proof-irrelevance, $\eta$-conversion and commuting conversions
% must all be considered in the definition of a suitable equivalence relation.

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% 
% \bibitem{FORMAVIE} The Formavie project:\\
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% 
% \bibitem{EHELM} The HELM project:\\
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% 
% \bibitem{MATHWEB} The MathWeb project:\\
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% 
% \bibitem{PCOQ} The PCoq project:\\
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% 
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\end{thebibliography}
 
\end{document}
